imported>WikiHarold: simplify

2024-02-05T11:56:34Z

simplify

New page

{{Short description|Lowest-energy state of a field in quantum field theories, corresponding to no particles present}}

[[File:Energy levels.svg|thumb|right|[[Physics:Energy level|Energy level]]s for an electron in an atom: ground state and excited states. In quantum field theory, the ground state is usually called the vacuum state or the vacuum.]]
{{Quantum field theory|cTopic=Topics}}

In [[Physics:Quantum field theory|quantum field theory]], the '''quantum vacuum state''' (also called the '''quantum vacuum''' or '''vacuum state''') is the [[Physics:Quantum state|quantum state]] with the lowest possible [[Physics:Energy|energy]]. Generally, it contains no physical particles. The term '''zero-point field''' is sometimes used as a synonym for the vacuum state of a quantized field which is completely individual.{{clarify|date=October 2023|reason=What does 'completely individual' mean?}}

According to present-day understanding of what is called the vacuum state or the quantum vacuum, it is "by no means a simple empty space".<ref name=Lambrecht>
{{cite book
|author=Astrid Lambrecht
|editor=Hartmut Figger
|editor2=Dieter Meschede
|editor3=Claus Zimmermann
|title=Observing mechanical dissipation in the quantum vacuum: an experimental challenge; in''' Laser physics at the limits'''
|page=197
|publisher= Springer
|location=Berlin/New York
|date=2002
|isbn=978-3-540-42418-5
|url=https://books.google.com/books?id=0DUjDAPwcqoC&q=%22vacuum+state%22&pg=PA197}}
</ref><ref name=Ray>
{{cite book
|author=Christopher Ray
|title=Time, space and philosophy
|page=Chapter 10, p. 205
|publisher= Routledge
|location=London/New York
|date=1991
|isbn=978-0-415-03221-6
|url=https://books.google.com/books?id=1F7xWULz0P0C&q=%22vacuum+state%22&pg=RA1-PA205
|no-pp=true}}
</ref> According to quantum mechanics, the vacuum state is not truly empty but instead contains fleeting [[Physics:Electromagnetic waves|electromagnetic waves]] and [[Physics:Particle|particle]]s that pop into and out of the quantum field.<ref>{{Cite web |url=http://www.aip.org/pnu/1996/split/pnu300-3.htm |title=AIP Physics News Update,1996 |access-date=2008-02-29 |archive-date=2008-01-29 |archive-url=https://web.archive.org/web/20080129093425/http://www.aip.org/pnu/1996/split/pnu300-3.htm |url-status=dead }}</ref><ref>[http://focus.aps.org/story/v2/st28 Physical Review Focus Dec. 1998]</ref><ref name=Dittrich>
{{cite book
|author=Walter Dittrich
|author2=Gies H
|name-list-style=amp
|title=Probing the quantum vacuum: perturbative effective action approach
|publisher= Springer
|location=Berlin
|date=2000
|isbn=978-3-540-67428-3
|url=https://archive.org/details/springer_10.1007-3-540-45585-X}}
</ref>

The [[Physics:QED vacuum|QED vacuum]] of [[Physics:Quantum electrodynamics|quantum electrodynamics]] (or QED) was the first vacuum of [[Physics:Quantum field theory|quantum field theory]] to be developed. QED originated in the 1930s, and in the late 1940s and early 1950s it was reformulated by [[Biography:Richard Feynman|Feynman]], Tomonaga, and [[Biography:Julian Schwinger|Schwinger]], who jointly received the Nobel prize for this work in 1965.<ref name=history>

For a historical discussion, see for example {{cite book |title=Historical Encyclopedia of Natural and Mathematical Sciences |volume=1 |editor=Ari Ben-Menaḥem |chapter-url=https://books.google.com/books?id=9tUrarQYhKMC&pg=PA4892 |chapter=Quantum electrodynamics (QED) |pages=4892 ''ff'' |isbn=978-3-540-68831-0 |date=2009 |publisher=Springer |edition=5th}} For the Nobel prize details and the Nobel lectures by these authors, see {{cite web |title=The Nobel Prize in Physics 1965 |url=https://www.nobelprize.org/nobel_prizes/physics/laureates/1965/ |publisher=Nobelprize.org |access-date=2012-02-06}}

</ref> Today the [[Physics:Electromagnetism|electromagnetic interaction]]s and the [[Physics:Weak interaction|weak interaction]]s are unified (at very high energies only) in the theory of the [[Physics:Electroweak interaction|electroweak interaction]].

The [[Physics:Standard Model|Standard Model]] is a generalization of the QED work to include all the known [[Physics:Elementary particle|elementary particle]]s and their interactions (except gravity). [[Physics:Quantum chromodynamics|Quantum chromodynamics]] (or QCD) is the portion of the Standard Model that deals with [[Physics:Strong interaction|strong interaction]]s, and [[Physics:QCD vacuum|QCD vacuum]] is the vacuum of quantum chromodynamics. It is the object of study in the [[Physics:Large Hadron Collider|Large Hadron Collider]] and the [[Physics:Relativistic Heavy Ion Collider|Relativistic Heavy Ion Collider]], and is related to the so-called vacuum structure of strong interactions.<ref name= Letessier>{{cite book |title=Hadrons and Quark-Gluon Plasma |author=Jean Letessier |author2=Johann Rafelski |page=37 ''ff'' |url=https://books.google.com/books?id=vSnFPyQaSTsC&q=weinberg+%22symmetry+%22&pg=PR11 |isbn=978-0-521-38536-7 |date=2002 |publisher=Cambridge University Press}}</ref>

==Non-zero expectation value==
{{main|Physics:Vacuum expectation value}}
[[File:Vacuum fluctuations revealed through spontaneous parametric down-conversion.ogv|thumb|right|350px| The video of an experiment showing vacuum fluctuations (in the red ring) amplified by spontaneous parametric down-conversion. ]]

If the quantum field theory can be accurately described through [[Perturbation theory (quantum mechanics)|perturbation theory]], then the properties of the vacuum are analogous to the properties of the [[Physics:Ground state|ground state]] of a quantum mechanical [[Physics:Harmonic oscillator|harmonic oscillator]], or more accurately, the [[Physics:Ground state|ground state]] of a [[Physics:Measurement problem|measurement problem]]. In this case the [[Physics:Vacuum expectation value|vacuum expectation value]] (VEV) of any [[Physics:Quantum field theory#Field operators|field operator]] vanishes. For quantum field theories in which perturbation theory breaks down at low energies (for example, [[Physics:Quantum chromodynamics|Quantum chromodynamics]] or the [[Physics:BCS theory|BCS theory]] of [[Physics:Superconductivity|superconductivity]]) field operators may have non-vanishing [[Physics:Vacuum expectation value|vacuum expectation value]]s called condensates. In the [[Physics:Standard Model|Standard Model]], the non-zero vacuum expectation value of the [[Physics:Higgs field|Higgs field]], arising from [[Physics:Spontaneous symmetry breaking|spontaneous symmetry breaking]], is the mechanism by which the other fields in the theory acquire mass.

==Energy==
{{main|Physics:Vacuum energy}}
The vacuum state is associated with a [[Physics:Zero-point energy|zero-point energy]], and this zero-point energy (equivalent to the lowest possible energy state) has measurable effects. In the laboratory, it may be detected as the [[Physics:Casimir effect|Casimir effect]]. In [[Astronomy:Physical cosmology|physical cosmology]], the energy of the cosmological vacuum appears as the [[Astronomy:Cosmological constant|cosmological constant]]. In fact, the energy of a cubic centimeter of empty space has been calculated figuratively to be one trillionth of an [[Physics:Erg|erg]] (or 0.6 eV).<ref>Sean Carroll, Sr Research Associate - Physics, [[Organization:California Institute of Technology|California Institute of Technology]], June 22, 2006 {{wipe|C-SPAN}} broadcast of Cosmology at Yearly Kos Science Panel, Part 1</ref> An outstanding requirement imposed on a potential Theory of Everything is that the energy of the quantum vacuum state must explain the physically observed cosmological constant.

==Symmetry==
For a [[Physics:Theory of relativity|relativistic]] field theory, the vacuum is Poincaré invariant, which follows from
[[Physics:Wightman axioms|Wightman axioms]] but can be also proved directly without these axioms.<ref name=proof-vac>{{cite journal|last=Bednorz|first=Adam|title=Relativistic invariance of the vacuum|journal=The European Physical Journal C|date=November 2013|volume=73|issue=12|pages=2654|doi=10.1140/epjc/s10052-013-2654-9|arxiv = 1209.0209 |bibcode = 2013EPJC...73.2654B |s2cid=39308527}}</ref> Poincaré invariance implies that only [[Physics:Scalar|scalar]] combinations of field operators have non-vanishing [[Physics:Vacuum expectation value|VEV's]]. The [[Physics:Vacuum expectation value|VEV]] may break some of the internal symmetries of the [[Physics:Lagrangian (field theory)|Lagrangian]] of the field theory. In this case the vacuum has less symmetry than the theory allows, and one says that [[Physics:Spontaneous symmetry breaking|spontaneous symmetry breaking]] has occurred. See [[Physics:Higgs mechanism|Higgs mechanism]], standard model.

==Non-linear permittivity==
{{main|Physics:Schwinger limit}}
Quantum corrections to Maxwell's equations are expected to result in a tiny nonlinear electric polarization term in the vacuum, resulting in a field-dependent electrical permittivity ε deviating from the nominal value ε0 of [[Physics:Vacuum permittivity|vacuum permittivity]].<ref name=Delphenich>{{cite arXiv|title=Nonlinear Electrodynamics and QED |author=David Delphenich |date=2006 |eprint=hep-th/0610088}}</ref> These theoretical developments are described, for example, in Dittrich and Gies.<ref name=Dittrich/>
The theory of [[Physics:Quantum electrodynamics|quantum electrodynamics]] predicts that the [[Physics:QED vacuum|QED vacuum]] should exhibit a slight [[Physics:Nonlinear optics|nonlinearity]] so that in the presence of a very strong electric field, the permitivity is increased by a tiny amount with respect to ε0. Subject to ongoing experimental efforts<ref>{{cite journal |last1=Battesti |first1=Rémy |last2=Beard |first2=Jerome |last3=Böser |first3=Sebastian |last4=Bruyant |first4=Nicolas |last5=Budker |first5=Dmitry |last6=Crooker |first6=Scott A. |last7=Daw |first7=Edward J. |last8=Flambaum |first8=Victor V. |last9=Inada |first9=Toshiaki |last10=Irastorza |first10=Igor G. |last11=Karbstein |first11=Felix |last12=Kim |first12=Dong Lak |last13=Kozlov |first13=Mikhail G. |last14=Melhem |first14=Ziad |last15=Phipps |first15=Arran |last16=Pugnat |first16=Pierre |last17=Rikken |first17=Geert |last18=Rizzo |first18=Carlo |last19=Schott |first19=Matthias |last20=Semertzidis |first20=Yannis K. |last21=ten Kate |first21=Herman H.J. |last22=Zavattini |first22=Guido |display-authors=1 |title=High magnetic fields for fundamental physics |journal=Physics Reports |date=November 2018 |volume=765-766 |pages=1–39 |doi=10.1016/j.physrep.2018.07.005|arxiv=1803.07547 |bibcode=2018PhR...765....1B |s2cid=4931745 }}</ref> is the possibility that a strong electric field would modify the effective permeability of free space, becoming anisotropic with a value slightly below ''μ''0 in the direction of the electric field and slightly exceeding ''μ''0 in the perpendicular direction. The quantum vacuum exposed to an electric field thereby exhibits [[Physics:Birefringence|birefringence]] for an electromagnetic wave travelling in a direction other than that of the electric field. The effect is similar to the [[Physics:Kerr effect|Kerr effect]] but without matter being present.<ref name=Mourou>Mourou, G. A., T. Tajima, and S. V. Bulanov, [http://link.aps.org/doi/10.1103/RevModPhys.78.309 ''Optics in the relativistic regime''; § XI ''Nonlinear QED''], ''Reviews of Modern Physics'' vol. '''78''' (no. 2), 309-371 (2006) [https://web.archive.org/web/20030928093337/http://acc-physics.kek.jp/sokensympo/frontier_accelerator/FACW1-PROC/FACW1-23.1.pdf pdf file].</ref> This tiny nonlinearity can be interpreted in terms of virtual [[Physics:Pair production#Photon to electron and positron|pair production]]<ref>Klein, James J. and B. P. Nigam, [http://prola.aps.org/abstract/PR/v135/i5B/pB1279_1 ''Birefringence of the vacuum''], ''Physical Review'' vol. '''135''', p. B1279-B1280 (1964).</ref> A characteristic electric field strength for which the nonlinearities become sizable is predicted to be enormous, about <math>1.32 \times 10^{18}</math>V/m, known as the [[Physics:Schwinger limit|Schwinger limit]]; the equivalent Kerr constant has been estimated, being about 1020 times smaller than the Kerr constant of water. Explanations for [[Physics:Dichroism|dichroism]] from particle physics, outside quantum electrodynamics, also have been proposed.<ref>{{cite journal |author1=Holger Gies |author2=Joerg Jaeckel |author3=Andreas Ringwald |doi=10.1103/PhysRevLett.97.140402 |date=2006 |title=Polarized Light Propagating in a Magnetic Field as a Probe of Millicharged Fermions |issue=14 |volume=97 |journal=Physical Review Letters |arxiv=hep-ph/0607118|bibcode = 2006PhRvL..97n0402G |pmid=17155223 |page=140402|s2cid=43654455 }}</ref> Experimentally measuring such an effect is very difficult,<ref>{{cite arXiv |eprint=0704.0748 |author1=Davis |author2=Joseph Harris |author3=Gammon |author4=Smolyaninov |author5=Kyuman Cho |title=Experimental Challenges Involved in Searches for Axion-Like Particles and Nonlinear Quantum Electrodynamic Effects by Sensitive Optical Techniques |class=hep-th |date=2007}}</ref> and has not yet been successful.

==Virtual particles==
{{main|Physics:Virtual particle}}
The presence of virtual particles can be rigorously based upon the [[Commutator|non-commutation]] of the [[Physics:Quantization of the electromagnetic field|quantized electromagnetic field]]s. Non-commutation means that although the [[Average|average]] values of the fields vanish in a quantum vacuum, their [[Variance|variances]] do not.<ref name=commutator>{{cite book |title=Modern nonlinear optics, Volume 85, Part 3|url=https://books.google.com/books?id=25LX8F2ybCsC&pg=PA462 |page=462 |quote=For all field states that have classical analog the field quadrature [[Variance|variances]] are also greater than or equal to this commutator. |isbn=978-0-471-57548-1 |publisher=John Wiley & Sons |date=1994 |author=Myron Wyn Evans |author2=Stanisław Kielich}}</ref> The term "[[Physics:Quantum fluctuation|vacuum fluctuation]]s" refers to the variance of the field strength in the minimal energy state,<ref name=Klyshko>{{cite book |title=Photons and nonlinear optics |author=David Nikolaevich Klyshko |url=https://books.google.com/books?id=IPfwdhR4TaYC&pg=PA126 |page=126 |isbn=978-2-88124-669-2|date=1988 |publisher=Taylor & Francis}}</ref> and is described picturesquely as evidence of "virtual particles".<ref name=Munitz>
{{cite book
|title=Cosmic Understanding: Philosophy and Science of the Universe
|url=https://books.google.com/books?id=HkOg14hXqi8C&pg=PA132.
|quote=The spontaneous, temporary emergence of particles from vacuum is called a "vacuum fluctuation".
|page=132
|isbn=978-0-691-02059-4
|author=Milton K. Munitz
|publisher=Princeton University Press
|date=1990}}
</ref> It is sometimes attempted to provide an intuitive picture of virtual particles, or variances, based upon the Heisenberg [[Physics:Uncertainty principle#Energy.E2.80.93time uncertainty principle|energy-time uncertainty principle]]:
<math display="block">\Delta E \Delta t \ge \frac{\hbar}{2} \, , </math>
(with Δ''E'' and Δ''t'' being the [[Physics:Energy|energy]] and [[Time|time]] variations respectively; Δ''E'' is the accuracy in the measurement of energy and Δ''t'' is the time taken in the measurement, and {{math|''ħ''}} is the [[Physics:Reduced Planck constant|Reduced Planck constant]]) arguing along the lines that the short lifetime of virtual particles allows the "borrowing" of large energies from the vacuum and thus permits particle generation for short times.<ref name=Davies/> Although the phenomenon of virtual particles is accepted, this interpretation of the energy-time uncertainty relation is not universal.<ref name=Allday/><ref name=King/> One issue is the use of an uncertainty relation limiting measurement accuracy as though a time uncertainty Δ''t'' determines a "budget" for borrowing energy Δ''E''. Another issue is the meaning of "time" in this relation, because energy and time (unlike position {{math|''q''}} and momentum {{math|''p''}}, for example) do not satisfy a [[Physics:Canonical commutation relation|canonical commutation relation]] (such as {{math|[''q'', ''p''] {{=}} i ''ħ''}}).<ref name=commutation/> Various schemes have been advanced to construct an observable that has some kind of time interpretation, and yet does satisfy a canonical commutation relation with energy.<ref name=Busch0/><ref name=Busch/> The very many approaches to the energy-time uncertainty principle are a long and continuing subject.<ref name=Busch/>

==Physical nature of the quantum vacuum==
According to Astrid Lambrecht (2002): "When one empties out a space of all matter and lowers the temperature to absolute zero, one produces in a ''Gedankenexperiment'' [thought experiment] the quantum vacuum state."<ref name=Lambrecht/> According to Fowler & [[Biography:Edward A. Guggenheim|Guggenheim]] (1939/1965), the [[Physics:Third law of thermodynamics|third law of thermodynamics]] may be precisely enunciated as follows:
<blockquote>It is impossible by any procedure, no matter how idealized, to reduce any assembly to the absolute zero in a finite number of operations.<ref>Fowler, R., [[Biography:Edward A. Guggenheim|Guggenheim, E.A.]] (1965). ''Statistical Thermodynamics. A Version of Statistical Mechanics for Students of Physics and Chemistry'', reprinted with corrections, Cambridge University Press, London, page 224.</ref> (See also.<ref>Partington, J.R. (1949). ''An Advanced Treatise on Physical Chemistry'', volume 1, ''Fundamental Principles. The Properties of Gases'', Longmans, Green and Co., London, page 220.</ref><ref>Wilks, J. (1971). The Third Law of Thermodynamics, Chapter 6 in ''Thermodynamics'', volume 1, ed. W. Jost, of H. Eyring, D. Henderson, W. Jost, ''Physical Chemistry. An Advanced Treatise'', Academic Press, New York, page 477.</ref><ref>Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics, New York, {{ISBN|0-88318-797-3}}, page 342.</ref>)</blockquote>

Photon-photon interaction can occur only through interaction with the vacuum state of some other field, for example through the Dirac electron-positron vacuum field; this is associated with the concept of [[Physics:Vacuum polarization|vacuum polarization]].<ref>Jauch, J.M., Rohrlich, F. (1955/1980). ''The Theory of Photons and Electrons. The Relativistic Quantum Field Theory of Charged Particles with Spin One-half'', second expanded edition, Springer-Verlag, New York, {{ISBN|0-387-07295-0}}, pages 287–288.</ref> According to [[Biography:Peter W. Milonni|Milonni]] (1994): "... all quantum fields have zero-point energies and vacuum fluctuations."<ref>Milonni, P.W. (1994). ''The Quantum Vacuum. An Introduction to Quantum Electrodynamics'', Academic Press, Inc., Boston, {{ISBN|0-12-498080-5}}, page xv.</ref> This means that there is a component of the quantum vacuum respectively for each component field (considered in the conceptual absence of the other fields), such as the electromagnetic field, the Dirac electron-positron field, and so on. According to Milonni (1994), some of the effects attributed to the [[Physics:QED vacuum|vacuum electromagnetic field]] can have several physical interpretations, some more conventional than others. The [[Physics:Casimir effect|Casimir attraction]] between uncharged conductive plates is often proposed as an example of an effect of the vacuum electromagnetic field. Schwinger, DeRaad, and Milton (1978) are cited by Milonni (1994) as validly, though unconventionally, explaining the Casimir effect with a model in which "the vacuum is regarded as truly a state with all physical properties equal to zero."<ref>Milonni, P.W. (1994). ''The Quantum Vacuum. An Introduction to Quantum Electrodynamics'', Academic Press, Inc., Boston, {{ISBN|0-12-498080-5}}, page 239.</ref><ref>{{cite journal | last1 = Schwinger | first1 = J. | last2 = DeRaad | first2 = L.L. | last3 = Milton | first3 = K.A. | year = 1978 | title = Casimir effect in dielectrics | journal = Annals of Physics | volume = 115 | issue = 1| pages = 1–23 | doi=10.1016/0003-4916(78)90172-0| bibcode = 1978AnPhy.115....1S }}</ref> In this model, the observed phenomena are explained as the effects of the electron motions on the electromagnetic field, called the source field effect. Milonni writes:

<blockquote>The basic idea here will be that the Casimir force may be derived from the source fields alone even in completely conventional QED, ... Milonni provides detailed argument that the measurable physical effects usually attributed to the vacuum electromagnetic field cannot be explained by that field alone, but require in addition a contribution from the self-energy of the electrons, or their radiation reaction. He writes: "The radiation reaction and the vacuum fields are two aspects of the same thing when it comes to physical interpretations of various QED processes including the [[Physics:Lamb shift|Lamb shift]], [[Physics:Van der Waals force|van der Waals force]]s, and Casimir effects."<ref>Milonni, P.W. (1994). ''The Quantum Vacuum. An Introduction to Quantum Electrodynamics'', Academic Press, Inc., Boston, {{ISBN|0-12-498080-5}}, page 418.</ref></blockquote>

This point of view is also stated by Jaffe (2005): "The Casimir force can be calculated without reference to vacuum fluctuations, and like all other observable effects in QED, it vanishes as the fine structure constant, {{math|''α''}}, goes to zero."<ref>Jaffe, R.L. (2005). Casimir effect and the quantum vacuum, ''Phys. Rev. D'' '''72''': 021301(R), http://1–5.cua.mit.edu/8.422_s07/jaffe2005_casimir.pdf{{Dead link|date=July 2018 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>

==Notations==
The vacuum state is written as <math>|0\rangle</math> or <math>|\rangle</math>. The [[Physics:Vacuum expectation value|vacuum expectation value]] (see also [[Physics:Expectation value (quantum mechanics)|Expectation value]]) of any field <math>\phi</math> should be written as <math>\langle0|\phi|0\rangle</math>.

==See also==
{{div col|colwidth=22em}}
* [[Physics:Pair production|Pair production]]
* [[Physics:Vacuum energy|Vacuum energy]]
* [[Physics:Lamb shift|Lamb shift]]
* [[Physics:False vacuum decay|False vacuum decay]]
* [[Physics:Squeezed coherent state|Squeezed coherent state]]
* [[Physics:Quantum fluctuation|Quantum fluctuation]]
* [[Physics:Scharnhorst effect|Scharnhorst effect]]
* [[Physics:Van der Waals force|Van der Waals force]]*
* [[Physics:Casimir effect|Casimir effect]]
{{div col end}}

==References and notes==
{{Reflist|refs=
<ref name=Allday>
A vaguer description is provided by {{cite book |title=Quarks, leptons and the big bang |author=Jonathan Allday |url=https://books.google.com/books?id=kgsBbv3-9xwC&pg=PA224 |pages=224 ''ff'' |quote=The interaction will last for a certain duration ''Δt''. This implies that the amplitude for the total energy involved in the interaction is spread over a range of energies ''ΔE''. |isbn=978-0-7503-0806-9 |edition=2nd |publisher=CRC Press |date=2002}}
</ref>

<ref name=Busch>
For a review, see {{cite book |title=Time in Quantum Mechanics |volume=734 |editor=J.G. Muga |editor2=R. Sala Mayato |editor3=Í.L. Egusquiza |pages=73–105 |chapter=Chapter 3: The Time–Energy Uncertainty Relation |author=Paul Busch |isbn=978-3-540-73472-7 |date=2008 |edition=2nd |publisher=Springer|arxiv=quant-ph/0105049 |bibcode=2002tqm..conf...69B |doi=10.1007/978-3-540-73473-4_3 |series=Lecture Notes in Physics |s2cid=14119708 }}
</ref>

<ref name=Busch0>
{{cite book |title=Operational quantum physics |url=https://archive.org/details/operationalquant00busc |url-access=limited |author=Paul Busch |author2=Marian Grabowski |author3=Pekka J. Lahti |chapter=§III.4: Energy and time |pages=[https://archive.org/details/operationalquant00busc/page/n86 77]''ff'' |isbn=978-3-540-59358-4 |date=1995 |publisher=Springer}}
</ref>

<ref name=commutation>
Quantities satisfying a canonical commutation rule are said to be noncompatible observables, by which is meant that they can both be measured simultaneously only with limited precision. See {{cite book |title=Encyclopedic dictionary of mathematics |author=Kiyosi Itô |chapter-url=https://books.google.com/books?id=azS2ktxrz3EC&pg=PA1303 |pages=1303 |chapter=§ 351 (XX.23) C: Canonical commutation relations |isbn=978-0-262-59020-4 |date=1993 |edition=2nd |publisher=MIT Press}}
</ref>

<ref name=Davies>
For an example, see {{cite book |title=The accidental universe |author=P. C. W. Davies |url=https://archive.org/details/accidentaluniver0000davi |url-access=registration |pages=[https://archive.org/details/accidentaluniver0000davi/page/106 106] |isbn=978-0-521-28692-3 |date=1982 |publisher=Cambridge University Press}}
</ref>

<ref name=King>
This "borrowing" idea has led to proposals for using the zero-point energy of vacuum as an infinite reservoir and a variety of "camps" about this interpretation. See, for example, {{cite book |title=Quest for zero point energy: engineering principles for 'free energy' inventions |author=Moray B. King |url=https://books.google.com/books?id=0RmkmrFxHM0C&pg=PA124 |pages=124 ''ff'' |isbn=978-0-932813-94-7 |date=2001 |publisher=Adventures Unlimited Press}}
</ref>

}}

==Further reading==
* Free pdf copy of [http://www.physics.arizona.edu/~rafelski/Books/StructVacuumE.pdf The Structured Vacuum - thinking about nothing] by [[Biography:Johann Rafelski|Johann Rafelski]] and Berndt Muller (1985) {{ISBN|3-87144-889-3}}.
* M.E. Peskin and D.V. Schroeder, ''An introduction to Quantum Field Theory''.
* H. Genz, '' Nothingness: The Science of Empty Space''
* {{cite arXiv |eprint=astro-ph/0107316|last1= Puthoff|first1= H. E.|title= Engineering the Zero-Point Field and Polarizable Vacuum for Interstellar Flight|last2= Little|first2= S. R.|last3= Ibison|first3= M.|year= 2001}}
* E. W. Davis, V. L. Teofilo, B. Haisch, H. E. Puthoff, L. J. Nickisch, A. Rueda and D. C. Cole(2006)"[http://www.calphysics.org/articles/Davis_STAIF06.pdf Review of Experimental Concepts for Studying the Quantum Vacuum Field]"

==External links==
* [https://web.archive.org/web/19980224172207/http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/970724a.html Energy into Matter]

{{Quantum mechanics topics}}

[[Category:Quantum field theory]]
[[Category:Vacuum]]
[[Category:Quantum states]]

{{Sourceattribution|Quantum vacuum state}}

← Older revision		Revision as of 11:56, 5 February 2024
(No difference)

Physics:Quantum vacuum state - Revision history

imported>WikiHarold: simplify

imported>WikiHarold: simplify